Trade Credit Policy Between Supplier–Manufacturer–Retailer for Ameliorating/Deteriorating Items

Abstract

This paper is related to the advancement of the inventory models for ameliorating items and focused on the real-life business situation as with the time the deterioration rate of ameliorating items is increased. In the global world, every supply chain entities as suppliers/manufacturers/retailers want to increase the consumption of their goods without any losses. For this, he/she tries to lure manufacturer/retailers by offering some discounts, i.e. credit period for settling the account. The problem states that the manufacturer purchases the ameliorating items from the supplier, where the supplier offers his/her credit period to settle the account. The manufacturer purchases ameliorating items (like pigs, fishes, ducklings, etc.) and take those items as raw material; when the livestock matures the manufacturer sells it to the retailer and offer credit time for settling the account. Reason to propose the model is when the quantities of livestock become larger, then the manufacturer faces difficulty in maintaining all the livestock. In such a situation, the traditional method (without offering credit period) fails to provide the maximum profit to the manufacturer. Therefore, in order to get maximum profit, the manufacturer needs some more realistic scientific outlook for making decisions. The proposed model provides a more realistic assumption of business markets, by offering credit policy. In the introduced model, manufacturer faces amelioration and deterioration rate simultaneously due to the growth and the death of livestock. The amelioration and deterioration rates are assumed as the Weibull distribution type. Shortages allowed only for the retailer, which is partially backlogged. The main goal of this paper is to minimize the total relevant inventory cost for both the manufacturer and the retailers, by finding the optimal replenishment policy. The mathematical formulation with optimal solutions for manufacturer and retailers are given. Convexity and existence of the proposed model via numerical examples and graphical representations are explained. Finally, the conclusions with some future research direction are discussed.

Appendix

1.1 Optimal Solution for Supplier

Case 1. When $M_\mathrm{manu} \leqslant T_1$

Since the equation for $TC_1$ is highly nonlinear, it is not possible to write here the whole expression, and we just give the brief solution procedure. First, we will take the Hessian matrix given below:

$$\begin{aligned} H_m(T_1, T_2) = \tfrac{\partial ^2 TC_1}{\partial T_1^2} \tfrac{\partial ^2 TC_1}{\partial T_2^2} - \bigg (\tfrac{\partial ^2 TC_1}{\partial T_1 \partial T_2}\bigg )^2 > 0. \end{aligned}$$

Because the value of Hessian matrix $H_m(T_1, T_2)$ is highly nonlinear, it is not possible to show that the direct existence of the equation. The Hessian matrix $H_m(T_1, T_2) > 0$ only if $y, \beta > 0$; $(\beta + y) > 0$; $(2 \beta + y) > 1$; $M_\mathrm{manu} \geqslant 0, T_2 \geqslant 0$; $\tfrac{M_\mathrm{manu}}{(T_2 - M_\mathrm{manu})} \geqslant 0$; $M_\mathrm{manu} \ne 0$; and $M_\mathrm{manu}^2 \ne M_\mathrm{manu} T_2$.

Case 2. When $T_1 \leqslant M_\mathrm{manu}$

For sufficient condition, we partially differentiate $TC_2(T_1, T_2)$ with respect to $T_1$ and $T_2$ and equate it to be zero. Thus, we have

$$\begin{aligned} {}&\tfrac{\partial TC_2}{\partial T_1} \nonumber \\ =&\tfrac{1}{{2 ({T_1}+{T_2})^2}} \biggl ({C_{\mathrm{ma}}} D_m {T_2} ({T_1}+{T_2}) x y \bigg ((a \beta {T_1}^\beta -{T_1}^y x y) \bigg (\tfrac{x {T_1}^y+2}{y} \nonumber \\&-\,\tfrac{2 \alpha {T_1}^\beta }{\beta +y}\bigg ) + y \bigg (-x {T_1}^y+\alpha ({T_1}^\beta +\tfrac{{T_2}^\beta }{\beta +1})-\tfrac{{T_2}^y x}{y+1}+1\bigg ) \bigg (\tfrac{x {T_1}^y+2}{y}-\tfrac{2 \alpha {T_1}^\beta }{\beta +y}\bigg ) \nonumber \\&\quad +\, \bigg (-x {T_1}^y+\alpha \bigg ({T_1}^\beta +\tfrac{{T_2}^\beta }{\beta +1}\bigg )-\tfrac{{T_2}^y x}{y+1}+1\bigg ) \bigg ({T_1}^y x-\tfrac{2 \alpha \beta {T_1}^\beta }{\beta +y}\bigg )\bigg ) {T_1}^{y-1} \nonumber \\&\quad -\, 2 {A_m} - 2 D_m {I_{me}} {S_m} ({M_\mathrm{manu}}-{T_1}-{T_2}) ({T_1}+{T_2}) + D_m {I_{m_e}} {S_m} (2 {M_\mathrm{manu}}-{T_1}-{T_2}) ({T_1}+{T_2})\nonumber \\&\quad - \,2 {C_m} D_m {T_2} \bigg (-x {T_1}^y+\alpha ({T_1}^\beta + \tfrac{{T_2}^\beta }{\beta +1})-\tfrac{{T_2}^y x}{y+1}+1\bigg ) \nonumber \\&\quad +\, 2 D_m {h_m} {T_2} ({T_1}+{T_2}) \bigg ((\alpha \beta {T_1}^\beta -{T_1}^y x y) \bigg (-\tfrac{\alpha {T_1}^\beta }{\beta +1}+\tfrac{x {T_1}^y}{y+1}+1\bigg )+\bigg (-x {T_1}^y \nonumber \\&\quad +\,\alpha \bigg ({T_1}^\beta +\tfrac{{T_2}^\beta }{\beta +1}\bigg )-\tfrac{{T_2}^y x}{y+1}+1\bigg ) \bigg (-\tfrac{\alpha {T_1}^\beta }{\beta +1}+\tfrac{x {T_1}^y}{y+1}+1\bigg )+\bigg (\tfrac{{T_1}^y x y}{y+1}\nonumber \\&\quad -\,\tfrac{\alpha \beta {T_1}^\beta }{\beta +1}\bigg ) \bigg (-x {T_1}^y+\alpha \bigg ({T_1}^\beta +\tfrac{{T_2}^\beta }{\beta +1}\bigg )-\tfrac{{T_2}^y x}{y+1}+1\bigg )\bigg )-D_m {h_m} {T_2} \nonumber \\&\quad \times \bigg (2 {T_1} \bigg (-\tfrac{\alpha {T_1}^\beta }{\beta +1}+\tfrac{x {T_1}^y}{y+1}+1\bigg ) \bigg (-x {T_1}^y+\alpha ({T_1}^\beta +\tfrac{{T_2}^\beta }{\beta +1})-\tfrac{{T_2}^y x}{y+1}+1\bigg ) \nonumber \\&\quad +\,{T_2} \bigg (\tfrac{2 \alpha ((\beta +2) x {T_2}^y+\beta (y+1)) {T_2}^\beta }{(\beta +1) (\beta +2) (y+1)}-\tfrac{\alpha ^2 {T_2}^{2 \beta }}{(\beta +1)^2}-\tfrac{x (x (y+2) {T_2}^y+2 y (y+1)) {T_2}^y}{(y+1)^2 (y+2)}+1\bigg )\bigg )\nonumber \\&\quad -\,{C_{m\alpha }} D_m {T_2} x y \bigg (\bigg (-x {T_1}^y+\alpha ({T_1}^\beta +\tfrac{{T_2}^\beta }{\beta +1})-\tfrac{{T_2}^y x}{y+1}+1\bigg ) \nonumber \\&\quad \bigg (\tfrac{x {T_1}^y+2}{y}-\tfrac{2 \alpha {T_1}^\beta }{\beta +y}\bigg ) {T_1}^y+{T_2}^y \bigg (\tfrac{\alpha (3 x y (y+1) {T_2}^y +\beta ^2 (x {T_2}^y+2)+\beta (x (4 y+1) {T_2}^y+4 y+2)) {T_2}^\beta }{y (\beta +y) (\beta +y+1) (\beta +2 y+1)}\nonumber \\&\quad -\,\tfrac{2 \alpha ^2 {T_2}^{2 \beta }}{(\beta +y) (2 \beta +y+1)}-\tfrac{2 x {T_2}^y}{2 y+1}-\tfrac{2 x {T_2}^y}{y^2+y}+\tfrac{2 x {T_2}^y}{2 y^2+3 y+1}-\,\tfrac{x^2 {T_2}^{2 y}}{y^2+y}+\tfrac{2 x^2 {T_2}^{2 y}}{3 y^2+4 y+1} \nonumber \\&\quad +\tfrac{x {T_2}^y+2}{y}-\tfrac{2}{y+1}\bigg )\bigg )+\tfrac{2 {C_m} D_m {T_2} ({T_1}+{T_2}) (\alpha \beta {T_1}^\beta -{T_1}^y x y)}{{T_1}} \biggl ) \end{aligned}$$

(7.1)

and

$$\begin{aligned} {}&\tfrac{\partial TC_2}{\partial T_2} = -\tfrac{1}{{2 ({T_1}+{T_2})^2}} \biggl ( 2 {A_m} + 2 D_m {I_{me}} {S_m} ({M_\mathrm{manu}}-{T_1}-{T_2}) ({T_1}+{T_2})-D_m {I_{m_e}} {S_m} (2 {M_\mathrm{manu}}-{T_1}-{T_2})\nonumber \\&\quad \times ({T_1}+{T_2})-2 {C_m} D_m ({T_1}+{T_2}) (-x {T_1}^y+\alpha ({T_1}^\beta +{T_2}^\beta )-{T_2}^y x+1)\nonumber \\&\quad +\,\tfrac{1}{{\beta +y}} \biggl ({C_{m\alpha }} D_m ({T_1}+{T_2}) x (\beta (-2 (\alpha {T_2}^\beta -x {T_2}^y+1) {T_1}^y + x (-\alpha {T_2}^\beta +x {T_2}^y+1) {T_1}^{2 y}\nonumber \\&\quad +\,x^2 {T_1}^{3 y}-2 \alpha {T_1}^{\beta +y}-\alpha x {T_1}^{\beta +2 y}+{T_2}^y (x {T_2}^y+2) (-\alpha {T_2}^\beta +x {T_2}^y-1))\nonumber \\&\quad +\,(-2 (\alpha {T_2}^\beta -x {T_2}^y+1) {T_1}^y+x (-\alpha {T_2}^\beta +x {T_2}^y+1) {T_1}^{2 y}+x^2 {T_1}^{3 y} + 2 \alpha (\alpha {T_2}^\beta \nonumber \\&\quad -\,{T_2}^y x) {T_1}^{\beta +y} + 2 \alpha ^2 {T_1}^{2 \beta +y}-3 \alpha x {T_1}^{\beta +2 y}+{T_2}^y (2 \alpha ^2 {T_2}^{2 \beta }+x {T_2}^y+x^2 {T_2}^{2 y}\nonumber \\&\quad -\,3 \alpha x {T_2}^{\beta +y}-2)) y) \biggl )+2 {C_m} D_m {T_2} \bigg (-x {T_1}^y+\alpha \bigg ({T_1}^\beta +\tfrac{{T_2}^\beta }{\beta +1}\bigg )-\tfrac{{T_2}^y x}{y+1}+1\bigg )\nonumber \\&\quad +\,D_m {h_m} {T_2} \bigg (2 {T_1} \bigg (-\tfrac{\alpha {T_1}^\beta }{\beta +1}+\tfrac{x {T_1}^y}{y+1}+1\bigg ) (-x {T_1}^y+\alpha ({T_1}^\beta +\tfrac{{T_2}^\beta }{\beta +1}\bigg )\nonumber \\&\quad -\tfrac{{T_2}^y x}{y+1}+1)+{T_2} \bigg (\tfrac{2 \alpha ((\beta +2) x {T_2}^y+\beta (y+1)) {T_2}^\beta }{(\beta +1) (\beta +2) (y+1)} -\tfrac{\alpha ^2 {T_2}^{2 \beta }}{(\beta +1)^2}-\tfrac{x (x (y+2) {T_2}^y+2 y (y+1)) {T_2}^y}{(y+1)^2 (y+2)}+1\bigg )\bigg )\nonumber \\&\quad -\,\tfrac{1}{{(\beta +1) (y+1)}} \bigl (2 D_m {h_m} ({T_1}+{T_2}) (\alpha (-\alpha {T_2}^\beta +x {T_2}^y+\beta ) (y+1) {T_1}^{\beta +1}\nonumber \\&\quad -\,\alpha ^2 (y+1) {T_1}^{2 \beta +1}-(\beta +1) x (-\alpha {T_2}^\beta +x {T_2}^y+y) {T_1}^{y+1}+\alpha x (\beta +y+2) {T_1}^{\beta +y+1}\nonumber \\&\quad -(\beta +1) x^2 {T_1}^{2 y+1}+(\beta +1) (\alpha {T_2}^\beta -x {T_2}^y+1) (y+1) {T_1}+{T_2} (\alpha {T_2}^\beta -x {T_2}^y+1)\nonumber \\&\quad (-\alpha (y+1) {T_2}^\beta +x {T_2}^y+y+\, \beta (x {T_2}^y+y+1)+1)) \bigl ) +{C_{m\alpha }} D_m {T_2} x y \nonumber \\&\quad \times \bigg (\bigg (-x {T_1}^y+\alpha ({T_1}^\beta +\tfrac{{T_2}^\beta }{\beta +1})-\tfrac{{T_2}^y x}{y+1}+1\bigg )\bigg (\tfrac{x {T_1}^y+2}{y}-\tfrac{2 \alpha {T_1}^\beta }{\beta +y}\bigg ) {T_1}^y\nonumber \\&\quad +{T_2}^y \bigg (\tfrac{\alpha (3 x y (y+1) {T_2}^y+\beta ^2 (x {T_2}^y+2)+\beta (x (4 y+1) {T_2}^y+4 y+2)) {T_2}^\beta }{y (\beta +y) (\beta +y+1) (\beta +2 y+1)}-\,\tfrac{2 \alpha ^2 {T_2}^{2 \beta }}{(\beta +y) (2 \beta +y+1)}-\tfrac{2 x {T_2}^y}{2 y+1}-\tfrac{2 x {T_2}^y}{y^2+y}\nonumber \\&\quad +\tfrac{2 x {T_2}^y}{2 y^2+3 y+1} -\tfrac{x^2 {T_2}^{2 y}}{y^2+y}+\tfrac{2 x^2 {T_2}^{2 y}}{3 y^2+4 y+1}+\tfrac{x {T_2}^y+2}{y}-\tfrac{2}{y+1}\bigg )\bigg ) \biggl ). \end{aligned}$$

(7.2)

Next, take the Hessian matrix with the help of above Eqs. (7.1) and (7.2) given below

$$\begin{aligned} H_m(T_1, T_2) = \tfrac{\partial ^2 TC_2}{\partial T_1^2} \tfrac{\partial ^2 TC_2}{\partial T_2^2} - \bigg (\tfrac{\partial ^2 TC_2}{\partial T_1 \partial T_2}\bigg )^2 > 0. \end{aligned}$$

Since the value of Hessian matrix is highly nonlinear, it is not possible to show the direct existence of the equation. The Hessian matrix exists, only if $y, \beta > 0$; $(\beta + y) > 0$; $(2 \beta + y) > 1$; ${M_\mathrm{manu}} \geqslant 0, T_2 \geqslant 0$; $\tfrac{{M_\mathrm{manu}}}{(T_2 - {M_\mathrm{manu}})} \geqslant 0$; ${M_\mathrm{manu}} \ne 0$; and ${M_\mathrm{manu}}^2 \ne {M_\mathrm{manu}} T_2$.

1.2 Optimal Solution for Retailer

Case 1. When $M_\mathrm{retail} \leqslant T_3$:

First, take the first-order partial differential equation of Eq. (3.31) with respect to $T_3$ and $T_4$ and equate to be zero, written as

$$\begin{aligned}&\tfrac{\partial TC_3}{\partial T_3} = \tfrac{D_r {I_{r_{p}}}}{2 ({y_1}+1)^2 ({y_1}+2) T_4} \biggl ( {T_3}^2 (2 {x_1} {y_1}^2 ({y_1}+1) {T_3}^{{y_1}-1}-2 {x_1}^2 {y_1} ({y_1}+2) {T_3}^{2 {y_1}-1})\nonumber \\&\quad +\,2 {x_1}^2 {y_1} ({y_1}+2) {M_\mathrm{retail}}^{{y_1}+1} {T_3}^{y_1} +2 {T_3} (-{x_1}^2 ({y_1}+2) {T_3}^{2 {y_1}}\nonumber \\&\quad +2 {x_1} {y_1} ({y_1}+1) {T_3}^{y_1}+({y_1}+1)^2 ({y_1}+2))+2 {x_1} ({y_1}+2) {M_\mathrm{retail}}^{{y_1}+1}\nonumber \\&\quad \times ({x_1} {T_3}^{y_1}+{y_1}+1)-2 {x_1} {y_1} ({y_1}+1) ({y_1}+2) {M_\mathrm{retail}} {T_3}^{y_1}-2 ({y_1}+1) ({y_1}+2)\nonumber \\&\quad \times {M_\mathrm{retail}} ({x_1} {T_3}^{y_1}+{y_1}+1) \biggl ) +\tfrac{{C_r} \left( D_r ({x_1} {T_3}^{y_1}+1)-\tfrac{D_r}{\delta (T_4-{T_3})+1}-D_r\right) }{T_4} \nonumber \\&\quad +\,\tfrac{D_r {h_r} {T_3}^2 \left( \tfrac{x_1 {y_1} {T_3}^{{y_1}-1} (2 {y_1} ({y_1}+1)-{x_1} ({y_1}+2) {T_3}^{y_1})}{({y_1}+1)^2 ({y_1}+2)}-\tfrac{{x_1}^2 {y_1} {T_3}^{2 {y_1}-1}}{({y_1}+1)^2}\right) }{2 T_4} \nonumber \\&\quad +\, \tfrac{D_r {h_r} {T_3} \left( \tfrac{{x_1} {T_3}^{y_1} (2 {y_1} ({y_1}+1)-{x_1} ({y_1}+2) {T_3}^{y_1})}{({y_1}+1)^2 ({y_1}+2)}+1\right) }{T_4} -\tfrac{{o_r} \left( 1-\tfrac{1}{\delta (T_4-{T_3})+1}\right) }{T_4} \nonumber \\&\quad -\, \tfrac{\delta {O_r} (T_4-{T_3})}{T_4 (\delta (T_4-{T_3})+1)^2}+\tfrac{{s_r} (\tfrac{\delta }{\delta (T_4-{T_3})+1}-\delta )}{\delta ^2 T_4}, \end{aligned}$$

(7.3)

$$\begin{aligned}&\tfrac{\partial TC_3}{\partial T_4} = -\tfrac{D_r {I_{r_{p}}}}{2 ({y_1}+1)^2 ({y_1}+2) T_4^2} (-{x_1}^2 ({y_1}+2) {M_\mathrm{retail}}^{2 {y_1}+2}+{T_3}^2 (-{x_1}^2 ({y_1}+2) {T_3}^{2 {y_1}} +2 {x_1} {y_1} ({y_1}+1) {T_3}^{y_1} \nonumber \\&\quad +\,({y_1}+1)^2 ({y_1}+2))+2 {x_1} ({y_1}+2) {T_3} {M_\mathrm{retail}}^{{y_1}+1} ({x_1} {T_3}^{y_1}+{y_1}+1)-2 {x_1} {y_1} ({y_1}+1) {M_\mathrm{retail}}^{{y_1}+2} \nonumber \\&\quad -\,2 ({y_1}+1) ({y_1}+2) {M_\mathrm{retail}} {T_3} ({x_1} {T_3}^{y_1}+{y_1}+1)+({y_1}+1)^2 ({y_1}+2) {M_\mathrm{retail}}^2) \nonumber \\&\quad -\,\tfrac{{C_r} (D_r (\tfrac{{x_1} {T_3}^{{y_1}+1}}{{y_1}+1} +{T_3})+\tfrac{D_r \log (\delta (T_4-{T_3})+1)}{\delta } - D_r{T_3})}{T_4^2} -\tfrac{D_r {h_r} {T_3}^2 (\tfrac{{x_1} {T_3}^{y_1} (2 {y_1} ({y_1}+1)-{x_1} ({y_1}+2) {T_3}^{y_1})}{({y_1}+1)^2 ({y_1}+2)}+1)}{2 T_4^2} \nonumber \\&\quad +\tfrac{{C_r} D_r}{T_4 (\delta (T_4-{T_3})+1)}+\tfrac{D_r {I_{r_{e}}} {M_\textit{retail}}^2 {S_p}}{2 T_4^2} -\tfrac{{O_r} (T_4-{T_3}) (1-\tfrac{1}{\delta (T_4-{T_3})+1})}{T_4^2}+\tfrac{{o_r} (1-\tfrac{1}{\delta (T_4-{T_3})+1})}{T_4} \nonumber \\&\quad +\,\tfrac{\delta {o_r} (T_4-{T_3})}{T_4 (\delta (T_4-{T_3})+1)^2}-\tfrac{{s_r} (\delta (T_4-{T_3})-\log (\delta (T_4-{T_3})+1))}{\delta ^2 T_4^2} +\tfrac{{s_r} (\delta -\tfrac{\delta }{\delta (T_4-{T_3})+1})}{\delta ^2 T_4}. \end{aligned}$$

(7.4)

For sufficient condition, we partially differentiate (4.1) with respect to $T_3$ and $T_4$ and check that the below Hessian condition exists

$$\begin{aligned} H_r(T_3, T_4) = \tfrac{\partial ^2 TC_3}{\partial T_3^2} \tfrac{\partial ^2 TC_3}{\partial T_4^2} - \left( \tfrac{\partial ^2 TC_3}{\partial T_3 \partial T_4}\right) ^2 > 0. \end{aligned}$$

For this, we take the second-order partial differential equation of $TC_3(T_3, T_4)$ and can be written as:

$$\begin{aligned} \tfrac{\partial ^2 TC_3}{\partial T_3^2}= & {} \tfrac{1}{{T_4}} \biggl ( D_r \bigg (\tfrac{{h_r} (-{x_1}^2 {T_3}^{2 {y_1}}-2 {x_1}^2 {y_1} {T_3}^{2 {y_1}}+{x_1} {y_1}^2 {T_3}^{y_1}+{x_1} {y_1} {T_3}^{y_1}+{y_1}+1)}{{y_1}+1} \nonumber \\&+\,\tfrac{{I_{r_{p}}} (-{x_1}^2 (2 {y_1}+1) {T_3}^{2 {y_1}+1} + {x_1} {y_1} {M_\text {retail}} {T_3}^{y_1} ({x_1} {M_\text {retail}}^{y_1}-{y_1}-1)+{x_1} {y_1} ({y_1}+1) {T_3}^{{y_1}+1}+({y_1}+1) {T_3})}{({y_1}+1) {T_3}}\nonumber \\&+\, {x_1} {y_1} {C_r} {T_3}^{{y_1}-1} - \tfrac{{C_r} \delta }{(\delta T_4-\delta {T_3}+1)^2}\bigg )+\tfrac{2 \delta {o_r}+\delta {s_r} T_4 - \delta {s_r} {T_3} + {s_r}}{(\delta T_4-\delta {T_3}+1)^3} \biggl ), \end{aligned}$$

(7.5)

$$\begin{aligned} \tfrac{\partial ^2 TC_3}{\partial T_4^2}= & {} \tfrac{1}{T_4^3} \biggl ( \tfrac{D_r {I_{r_{p}}}}{({y_1}+1)^2 ({y_1}+2)} \biggl (-{x_1}^2 ({y_1}+2) {M_\text {retail}}^{2 {y_1}+2} +{T_3}^2 (-{x_1}^2 ({y_1}+2) {T_3}^{2 {y_1}}\nonumber \\&+\,2 {x_1} {y_1} ({y_1}+1) {T_3}^{y_1} +({y_1}+1)^2 ({y_1}+2))+2 {x_1} ({y_1}+2) {T_3} {M_\text {retail}}^{{y_1}+1} ( {x_1} {T_3}^{y_1}+{y_1}+1)\nonumber \\&-\,2 {x_1} {y_1} ({y_1}+1) {M_\text {retail}}^{{y_1}+2} -2 ({y_1}+1) ({y_1}+2) {M_\text {retail}} {T_3} ({x_1} {T_3}^{y_1}+{y_1}+1)\nonumber \\&+\,({y_1}+1)^2 ({y_1}+2) {M_\text {retail}}^2 \bigg ) + \tfrac{2 {C_r} D_r ({x_1} \delta {T_3}^{{y_1}+1}+({y_1}+1) \log (\delta T_4-\delta {T_3}+1))}{({y_1}+1) \delta } \nonumber \\&+\, D_r {h_r} {T_3}^2 \bigg (\tfrac{{x_1} {T_3}^{y_1} (2 {y_1} ({y_1}+1)-{x_1} ({y_1}+2) {T_3}^{y_1})}{({y_1}+1)^2 ({y_1}+2)}+1\bigg )\nonumber \\&-\tfrac{{C_r} D_r \delta T_4^2}{(\delta T_4-\delta {T_3}+1)^2}-\tfrac{2 {C_r} D_r T_4}{\delta T_4-\delta {T_3}+1}- D_r {I_{r_{e}}} {M_\text {retail}}^2 {S_p}\nonumber \\&-\,\tfrac{2 \delta ^2 {o_r} T_4^2 (T_4-{T_3})}{(\delta T_4-\delta {T_3}+1)^3}+\tfrac{2 \delta {o_r} T_4^2}{(\delta T_4-\delta {T_3}+1)^2} - 2 {o_r} T_4 \bigg (\tfrac{1}{\delta (-T_4)+\delta {T_3}-1}+1\bigg ) \nonumber \\&+\, \tfrac{2 \delta {o_r} (T_4-{T_3})^2}{\delta T_4-\delta {T_3}+1}-\tfrac{2 \delta {o_r} T_4 (T_4-{T_3})}{(\delta T_4-\delta {T_3}+1)^2} + \tfrac{{s_r} T_4^2}{(\delta T_4-\delta {T_3}+1)^2} + \tfrac{2 {s_r} (\delta (T_4-{T_3})-\log (\delta T_4-\delta {T_3}+1))}{\delta ^2} \nonumber \\&-\,\tfrac{2 {s_r} T_4 (T_4-{T_3})}{\delta T_4-\delta {T_3}+1} \biggl ), \end{aligned}$$

(7.6)

$$\begin{aligned} \tfrac{\partial ^2 TC_3}{\partial T_3 \partial T_4}= & {} \tfrac{1}{T_4^2} \biggl (-\tfrac{{x_1} {y_1} D_r {h_r} {T_3}^{{y_1}+1} (-2 {x_1} {T_3}^{y_1}-{x_1} {y_1} {T_3}^{y_1}+{y_1}^2+{y_1})}{({y_1}+1)^2 ({y_1}+2)} +{C_r} D_r (\tfrac{1}{\delta T_4-\delta {T_3}+1}-{x_1} {T_3}^{y_1}) \nonumber \\&-\, D_r {h_r} {T_3} \bigg (\tfrac{{x_1} {T_3}^{y_1} (2 {y_1} ({y_1}+1)-{x_1} ({y_1}+2) {T_3}^{y_1})}{({y_1}+1)^2 ({y_1}+2)}+1\bigg ) + \tfrac{{C_r} D_r \delta T_4}{(\delta T_4-\delta {T_3}+1)^2} + \tfrac{2 \delta ^2 {o_r} T_4 (T_4-{T_3})}{(\delta T_4-\delta {T_3}+1)^3} \nonumber \\&+\, \tfrac{D_r {I_{r_{p}}} ({x_1} {T_3}^{y_1}+1) (-{x_1} {M_\text {retail}}^{{y_1}+1}-{T_3} (-{x_1} {T_3}^{y_1}+{y_1}+1)+({y_1}+1) {M_\text {retail}})}{{y_1}+1} \nonumber \\&+\, {o_r} \bigg (\tfrac{1}{\delta (-T_4)+\delta {T_3}-1}+1\bigg ) - \tfrac{2 \delta {o_r} T_4}{(\delta T_4-\delta {T_3}+1)^2} \nonumber \\&+\, \tfrac{\delta {o_r} (T_4-{T_3})}{(\delta T_4-\delta {T_3}+1)^2} +\tfrac{{s_r} (T_4-{T_3})}{\delta T_4-\delta {T_3}+1}-\tfrac{{s_r} T_4}{(\delta T_4-\delta {T_3}+1)^2} \biggl ). \end{aligned}$$

(7.7)

Since the value of Hessian matrix is highly nonlinear, it is not possible to prove the direct existence of the equation. The Hessian matrix exists only if $M_\mathrm{retail} \geqslant 0$, $T_3 \geqslant 0$, $\tfrac{M_\mathrm{retail}}{T_3 - M_\mathrm{retail}} \geqslant 0$, ${M_\mathrm{retail}}^2 \ne M_\mathrm{retail} \ \ T_3$, Re$(y_1) > -1$, and $\tfrac{M_\mathrm{retail}}{(M_\mathrm{retail} - T_3)} > 1$.

Case (2) $T_3 \leqslant M_\text {retail}$:

Similarly, take the first-order partial differential equation of Eq. (3.32) with respect to $T_3$ and $T_4$ and equate to be zero, written as

$$\begin{aligned} \tfrac{\partial TC_4}{\partial T_3}= & {} \tfrac{1}{{({T_3}+{T_4})^3}} \biggl ( \tfrac{2 x_1 y_1 D_r {h_r} {T_3}^{y_1+1} ({T_3}+{T_4}) (-2 x_1 {T_3}^{y_1}-x_1 y_1 {T_3}^{y_1}+y_1^2+y_1)}{(y_1+1)^2 (y_1+2)} \nonumber \\&-\, \tfrac{2 {C_r} D_r (x_1 \delta {T_3}^{y_1+1}+(y_1+1) \log (\delta (-{T_3})+\delta {T_4}+1))}{(y_1+1) \delta } + 2 {C_r} D_r ({T_3}+{T_4}) (x_1 {T_3}^{y_1} \nonumber \\&+\, \tfrac{1}{\delta {T_3}-\delta {T_4}-1}\bigg ) + 2 D_r {h_r} {T_3} ({T_3}+{T_4}) \left( \tfrac{x_1 {T_3}^{y_1} (2 y_1 (y_1+1)-x_1 (y_1+2) {T_3}^{y_1})}{(y_1+1)^2 (y_1+2)}+1\right) \nonumber \\&-\, D_r {h_r} {T_3}^2 \bigg (\tfrac{x_1 {T_3}^{y_1} (2 y_1 (y_1+1)-x_1 (y_1+2) {T_3}^{y_1})}{(y_1+1)^2 (y_1+2)} + 1\bigg ) - 2 {A_r} \nonumber \\&+\, D_r {I_{r_{e}}} {S_p} ({M_\text {retail}}^2+2 {M_\text {retail}} {T_3}-2 {T_3}^2)-2 D_r {I_{r_{e}}} {S_p}\nonumber \\&\times ({M_\text {retail}}-2 {T_3}) ({T_3}+{T_4}) - 2 {o_r} ({T_3}+{T_4})\nonumber \\&\times \bigg (\tfrac{1}{\delta {T_3}-\delta {T_4}-1}+1\bigg )+\tfrac{2 \delta {o_r} ({T_3}-{T_4}) ({T_3}+{T_4})}{(\delta (-{T_3})+\delta {T_4}+1)^2} + \tfrac{2 \delta {o_r} ({T_3}-{T_4})^2}{\delta {T_3}-\delta {T_4}-1} \nonumber \\&+\, \tfrac{2 {s_r} (\delta ({T_3}-{T_4})+\log (\delta (-{T_3})+\delta {T_4}+1))}{\delta ^2} -\tfrac{2 {s_r} ({T_3}-{T_4}) ({T_3}+{T_4})}{\delta {T_3}-\delta {T_4}-1} \biggl ). \end{aligned}$$

(7.8)

$$\begin{aligned} \tfrac{\partial TC_4}{\partial T_4}= & {} \tfrac{1}{{({T_3}+{T_4})^3}} \biggl (-\tfrac{2 {C_r} D_r (x_1 \delta {T_3}^{y_1+1}+(y_1+1) \log (\delta (-{T_3})+\delta {T_4}+1))}{(y_1+1) \delta } \nonumber \\&-\,D_r {h_r} {T_3}^2 \bigg (\tfrac{x_1 {T_3}^{y_1} (2 y_1 (y_1+1)-x_1 (y_1+2) {T_3}^{y_1})}{(y_1+1)^2 (y_1+2)}+1\bigg )-2 {A_r}+\tfrac{2 {C_r} D_r ({T_3}+{T_4})}{\delta (-{T_3})+\delta {T_4}+1} \nonumber \\&+\, D_r {I_{r_{e}}} {S_p} ({M_\text {retail}}^2+2 {M_\text {retail}} {T_3}-2 {T_3}^2)+2 {o_r} ({T_3}+{T_4}) \left( \tfrac{1}{\delta {T_3}-\delta {T_4}-1}+1\right) \nonumber \\&+\, \tfrac{2 \delta {o_r} ({T_4}-{T_3}) ({T_3}+{T_4})}{(\delta (-{T_3})+\delta {T_4}+1)^2} + \tfrac{2 \delta {o_r} ({T_3}-{T_4})^2}{\delta {T_3}-\delta {T_4}-1} + \tfrac{2 {s_r} (\delta ({T_3}-{T_4})+\log (\delta (-{T_3})+\delta {T_4}+1))}{\delta ^2} \nonumber \\&+\,\tfrac{2 {s_r} ({T_3}-{T_4}) ({T_3}+{T_4})}{\delta {T_3}-\delta {T_4}-1}\biggl ). \end{aligned}$$

(7.9)

For sufficient condition, we partially differentiate (3.32) with respect to $T_3$ and $T_4$ and check that the below Hessian condition exists:

$$\begin{aligned} H_{r_2}(T_3, T_4) = \tfrac{\partial ^2 TC_4}{\partial T_3^2} \tfrac{\partial ^2 TC_4}{\partial T_4^2} - \bigg (\tfrac{\partial ^2 TC_4}{\partial T_3 \partial T_4}\bigg )^2 > 0. \end{aligned}$$

For this, we take the second-order partial differential equation of Eq. (3.32), which is given below

$$\begin{aligned} \tfrac{\partial ^2 TC_4}{\partial T_3^2}= & {} \tfrac{1}{{({T_3}+{T_4})^3}} \biggl (-\tfrac{x_1 y_1 D_r {h_r} {T_3}^{y_1 + 1} ({T_3}+{T_4}) (-2 x_1 {T_3}^{y_1}-x_1 y_1 {T_3}^{y_1}+y_1^2+y_1)}{(y_1+1)^2 (y_1+2)} \nonumber \\&+\, \tfrac{2 {C_r} D_r (x_1 \delta {T_3}^{y_1+1}+(y_1+1) \log (\delta (-{T_3})+\delta {T_4}+1))}{(y_1+1) \delta } - {C_r} D_r ({T_3}+{T_4}) ( x_1 {T_3}^{y_1} \nonumber \\&+\,\tfrac{1}{\delta {T_3}-\delta {T_4}-1}\bigg ) - D_r {h_r} {T_3} ({T_3}+{T_4}) \left( \tfrac{x_1 {T_3}^{y_1} (2 y_1 (y_1+1)-x_1 (y_1+2) {T_3}^{y_1})}{(y_1+1)^2 (y_1+2)}+1\right) \nonumber \\&+\, D_r {h_r} {T_3}^2 \left( \tfrac{x_1 {T_3}^{y_1} (2 y_1 (y_1+1)-x_1 (y_1+2) {T_3}^{y_1})}{(y_1+1)^2 (y_1+2)}+1\right) + 2 {A_r} \nonumber \\&+\, \tfrac{{C_r} D_r \delta ({T_3}+{T_4})^2}{(\delta (-{T_3})+\delta {T_4}+1)^2} + \tfrac{{C_r} D_r ({T_3}+{T_4})}{\delta {T_3}-\delta {T_4}-1} -\, D_r {I_{r_{e}}} {S_p} ({M_\text {retail}}^2+2 {M_\text {retail}} {T_3}-2 {T_3}^2) \nonumber \\&+ D_r {I_{r_{e}}} {S_p} ({M_\mathrm{retail}} - 2 {T_3}) ({T_3}+{T_4}) +\, \tfrac{2 \delta ^2 {o_r} ({T_4}-{T_3}) ({T_3}+{T_4})^2}{(\delta (-{T_3})+\delta {T_4}+1)^3}\nonumber \\&-\tfrac{2 \delta {o_r} ({T_3}+{T_4})^2}{(\delta (-{T_3})+\delta {T_4}+1)^2} -\, \tfrac{2 \delta {o_r} ({T_3}-{T_4})^2}{\delta {T_3}-\delta {T_4}-1}\nonumber \\&-\tfrac{2 {s_r} (\delta ({T_3}-{T_4})+\log (\delta (-{T_3})+\delta {T_4}+1))}{\delta ^2} -\tfrac{{s_r} ({T_3}+{T_4})^2}{(\delta (-{T_3})+\delta {T_4}+1)^2} \biggl ). \end{aligned}$$

(7.10)

$$\begin{aligned} \tfrac{\partial ^2 TC_4}{\partial T_4^2}= & {} \tfrac{1}{{({T_3}+{T_4})^3}} \biggl (\tfrac{2 {C_r} D_r (x_1 \delta {T_3}^{y_1+1}+(y_1+1) \log (\delta (-{T_3})+\delta {T_4}+1))}{(y_1+1) \delta } \nonumber \\&+\, D_r {h_r} {T_3}^2 \bigg (\tfrac{x_1 {T_3}^{y_1} (2 y_1 (y_1+1)-x_1 (y_1+2) {T_3}^{y_1})}{(y_1+1)^2 (y_1+2)}+1\bigg ) + 2 {A_r} \nonumber \\&-\,\tfrac{{C_r} D_r \delta ({T_3}+{T_4})^2}{(\delta (-{T_3})+\delta {T_4}+1)^2}+\tfrac{2 {C_r} D_r ({T_3}+{T_4})}{\delta {T_3}-\delta {T_4}-1} \nonumber \\&-\, D_r {I_{r_{e}}} {S_p} ({M_\text {retail}}^2+2 {M_\text {retail}} {T_3}-2 {T_3}^2)-\tfrac{2 \delta ^2 {o_r} ({T_4}-{T_3}) ({T_3}+{T_4})^2}{(\delta (-{T_3})+\delta {T_4}+1)^3} \nonumber \\&-\, 2 {o_r} ({T_3}+{T_4}) \bigg (\tfrac{1}{\delta {T_3}-\delta {T_4}-1} + 1\bigg ) + \tfrac{2 \delta {o_r} ({T_3}+{T_4})^2}{(\delta (-{T_3})+\delta {T_4}+1)^2} \nonumber \\&+\, \tfrac{2 \delta {o_r} ({T_3}-{T_4}) ({T_3}+{T_4})}{(\delta (-{T_3})+\delta {T_4}+1)^2}-\tfrac{2 \delta {o_r} ({T_3}-{T_4})^2}{\delta {T_3}-\delta {T_4}-1} - \tfrac{2 {s_r} (\delta ({T_3}-{T_4})+\log (\delta (-{T_3})+\delta {T_4}+1))}{\delta ^2}\nonumber \\&+\tfrac{{s_r} ({T_3}+{T_4})^2}{(\delta (-{T_3})+\delta {T_4}+1)^2} - \tfrac{2 {s_r} ({T_3}-{T_4}) ({T_3}+{T_4})}{\delta {T_3}-\delta {T_4}-1} \biggl ). \end{aligned}$$

(7.11)

$$\begin{aligned} \tfrac{\partial ^2 TC_4}{\partial T_4^2}= & {} \tfrac{1}{{({T_3}+{T_4})^3}} \biggl (\tfrac{4 x_1 y_1 D_r {h_r} {T_3}^{y_1} ({T_3}+{T_4})^2 (-2 x_1 {T_3}^{y_1}-x_1 y_1 {T_3}^{y_1}+y_1^2+y_1)}{(y_1+1)^2 (y_1+2)} \nonumber \\&-\, \tfrac{2 x_1 y_1 D_r {h_r} {T_3}^{y_1+1} ({T_3}+{T_4}) (-2 x_1 {T_3}^{y_1}-x_1 y_1 {T_3}^{y_1}+y_1^2+y_1)}{(y_1+1)^2 (y_1+2)}\nonumber \\&-\, \tfrac{x_1 y_1 D_r {h_r} {T_3}^{y_1} ({T_3}+{T_4})^2 (2 x_1 y_1^2 {T_3}^{y_1}-2 x_1 {T_3}^{y_1}+3 x_1 y_1 {T_3}^{y_1}-y_1^3+y_1)}{(y_1+1)^2 (y_1+2)} \nonumber \\&+\,{C_r} ({T_3}+{T_4})^2 \left( x_1 y_1 D_r {T_3}^{y_1-1}-\tfrac{D_r \delta }{(\delta (-{T_3})+\delta {T_4}+1)^2}\right) \nonumber \\&+\, \tfrac{2 {C_r} D_r (x_1 \delta {T_3}^{y_1+1}+(y_1+1) \log (\delta (-{T_3})+\delta {T_4}+1))}{(y_1+1) \delta } \nonumber \\&-\, 2 {C_r} D_r ({T_3}+{T_4}) \left( x_1 {T_3}^{y_1} + \tfrac{1}{\delta {T_3}-\delta {T_4}-1}\right) - 2 D_r {h_r} {T_3} ({T_3}+{T_4}) \nonumber \\&\times \left( \tfrac{x_1 {T_3}^{y_1} (2 y_1 (y_1+1)-x_1 (y_1+2) {T_3}^{y_1})}{(y_1+1)^2 (y_1+2)}+1\right) + D_r {h_r} ({T_3}+{T_4})^2 \nonumber \\&\times \left( \tfrac{x_1 {T_3}^{y_1} (2 y_1 (y_1+1)-x_1 (y_1+2) {T_3}^{y_1})}{(y_1+1)^2 (y_1+2)}+1\right) \nonumber \\&+\, D_r {h_r} {T_3}^2 \bigg (\tfrac{x_1 {T_3}^{y_1} (2 y_1 (y_1+1)-x_1 (y_1+2) {T_3}^{y_1})}{(y_1+1)^2 (y_1+2)}+1\bigg ) \nonumber \\&+\, 2 {x_1r} - D_r {I_{r_{e}}} {S_p} ({M_\text {retail}}^2+2 {M_\text {retail}} {T_3}-2 {T_3}^2)+2 D_r {I_{r_{e}}} {S_p} \nonumber \\&\times ({M_\text {retail}}-2 {T_3}) ({T_3}+{T_4})+2 D_r {I_{re}} {S_p} ({T_3}+{T_4})^2-\tfrac{2 \delta ^2 {o_r} ({T_4}-{T_3}) ({T_3}+{T_4})^2}{(\delta (-{T_3})+\delta {T_4}+1)^3} \nonumber \\&+\, 2 {o_r} ({T_3}+{T_4}) (\tfrac{1}{\delta {T_3}-\delta {T_4}-1}+1) + \tfrac{2 \delta {o_r} ({T_3}+{T_4})^2}{(\delta (-{T_3})+\delta {T_4}+1)^2} \nonumber \\&+\, \tfrac{2 \delta {o_r} ({T_4}-{T_3}) ({T_3}+{T_4})}{(\delta (-{T_3})+\delta {T_4}+1)^2} - \tfrac{2 \delta {o_r} ({T_3}-{T_4})^2}{\delta {T_3}-\delta {T_4}-1} - \tfrac{2 {s_r} (\delta ({T_3}-{T_4})+\log (\delta (-{T_3})+\delta {T_4}+1))}{\delta ^2}\nonumber \\&+ \tfrac{{s_r} ({T_3}+{T_4})^2}{(\delta (-{T_3})+\delta {T_4}+1)^2} + \tfrac{2 {s_r} ({T_3}-{T_4}) ({T_3}+{T_4})}{\delta {T_3}-\delta {T_4}-1} \biggl ). \end{aligned}$$

(7.12)

Similarly, the Hessian matrix $(H_{r_2}(T_3, T_4))$ is highly nonlinear, and thus it is not possible to prove the direct existence of the equation. The Hessian matrix exists only if $M_\text {retail} \geqslant 0$, $T_3 \geqslant 0$, $\tfrac{M_\text {retail}}{T_3 - M_\text {retail}} \geqslant 0$, ${M_\text {retail}}^2 \ne M_\text {retail} \ \ T_3$, Re$(y_1) > - 1$, and $\tfrac{M_\text {retail}}{(M_\text {retail} - T_3)} > 1$.